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%\vglue 5pt

\centerline{\cmrX About The Twisted } 
\centerline{\cmrX Scherk Saddle Towers} 
%\smallskip
%\centerline{\cmrX }
%\centerline{H. Karcher, Version Oct. 2004}

\lf
The {\it Twisted Scherk Saddle Towers} are minimal surfaces that were
found in 1988 as deformations of the Scherk Saddle Towers. Therefore
one should first  look at these latter simpler surfaces. One can imagine
that one grips such a saddle tower at the top and the bottom and deforms
the surface by twisting it. Of course it is not clear whether this can
be done in such a way that the deformations stay minimal. 

\noindent
Fortunately, the most symmetric
Scherk Saddle Towers carry straight lines through their saddles, and it is
easy to imagine these lines staying on the twisted surface
and remaining as lines of symmetry. Using these lines we can obtain the
existence of the desired surfaces by solving the following Plateau Problem.

\noindent
Consider a pair of adjacent half-lines through one saddle and another
pair of half-lines, starting from the saddle above the first and
with one of its half-lines vertically above the sector between the
first two. These two ``broken lines'' cut a simply connected
strip out of the surface. (One can see it by selecting 
``Don't Show Reflections'' from the Action Menu) .

\noindent
Now, vice versa, start with the two broken-lines and solve the Plateau 
problem for this infinite boundary to find a minimal strip they bound. 
Finally use $180^\circ$ rotations
around the half-lines to extend the Plateau strip to a complete minimal
surface, a Twisted Scherk Saddle Tower. 
These surfaces played an important role 
in the development of the theory of minimal surfaces 
since---except  for the Helicoid itself---they were the first examples 
having helicoidal ends, 
\LF
The integer parameter ee controls the dihedral rotational symmetry of the
surface: the angle of each pair of half lines above is $\pi/ee$. The
parameter $aa$ controls the amount of twist, with  $aa=0$ giving the straight Scherk
Saddle Towers. We must keep $aa  < \pi/ee$ , since otherwise the existence 
construction fails. Of course the default morph varies aa.
\LF
In 3D-XplorMath minimal surfaces are computed via their Weierstra\ss\
representation. The Scherk Saddle Towers are parametrized by punctured
spheres and our twist deformation does not change this conformal type.
However, due to this twist, the Gau\ss\ map of the 
surface is not single-valued: it is a rather a multivalued function on the punctured
sphere, and this makes the computation more difficult than for the other
spherical minimal surfaces, since during the integration of the multivalued
Weierstra\ss\ integrand, it is necessary to use ana\-lytic continuation 
in order to guarantee
that we always have the correct value of the Gau\ss\ map. 

\noindent
The Weierstra\ss\ representation is given in:

\noindent
\noblackboxes
Karcher, H., \hskip 4pt Embedded Minimal Surfaces derived from 
Scherk's Examples,  Manuscripta math. 62 (1988), pp. 83 - 114.


\noindent
Recently Traizet and Weber have found a new construction of embedded
singly periodic minimal surfaces that can be illustrated with the
twisted Scherk saddle towers. Choose $aa$ close to its theoretical
limit $1/ee$ and try to see the resulting surfaces as made out of $ee$
ordinary helicoids. Obviously one has to allow modifications of the
helicoids in the middle, but away from the middle one can see these
helicoids well. Traizet and Weber were able to turn this observation
around. They found conditions how to place a collection of helicoids
so that they could prove the existence of a family of singly periodic
embedded min\-imal surfaces which converged to the given helicoids in
the same sense as the twisted Scherk saddle towers converge as
$aa \to \pm 1/ee$.

\vglue -3pt
\noindent
 H.K.



\bye
